Problem Set 3--solutions to additional problems

Do Problems 13.4,  13.6, 13.13, 13.14 in Atkins plus these additional problems.

Problem 1

a.)  Working through "Justification 13.5" in Atkins, verify the dipole allowed selection rules for a hydrogenic atom.

b Calculate the transition dipole moment for the 2p -> 1s,  3p -> 1s, and 4p -> 1s  transitions in the hydrogen atom by evaluating Eq. [28]. How does the intensity of the spectral lines change with increasing n?  For these examples, consider only the z-component (by symmetry the x & y components should be the same).

2p -> 1s

Here is the dipole operator:

The angular parts are the same in each case:

The radial parts are different

The intensities are proportional to the matrix elements squared.  So, we see that the intensity decreases with increasing quantum number.  For example, the ratio of the 2->1 vs 3->1 transition is roughly 10, i.e. the 2-> 1 transition is ~10⨯ more intense than the 3->1 transition.

Problem 2

In this problem we will work through how the variational principle works as applied to determining the energy levels of atoms.

Consider a two electron atom (such as He).  The effective electronic potential energy function for electron 1 in the presence of electron 2 is computed as
(I think the expression given in A&M may be in error —!, )

(where does this expression come from ??)

  We will use as a parameter in what follows.

a   Use this to construct the effective Hamiltonian for electron 1 as a function of .

Answer: The effective Hamiltonian for electron 1 is

b. Solve this one electron Schrodinger equation using the variational principle using the normalized trial function

taking ζ as a variational parameter.   To do this, take the Hamiltonian you derived in a, and compute the energy expectation value ε(ζ) =  ⟨H⟩ as a function of the variational parameter ζ..
For now, let = taking Z = 2 for the He atom and as the Bohr radius,  Plot ε(ζ) as a function of ζ Then, take the derivative of ε(ζ )

to determine the ζ which minimizes the energy.

Answer:   To do this part, we first construct H as a function of ζ (done in part a) and take the energy expectation value with respect to the trial wavefunction;  the kinetic part is easy, we just have to remember that the derivative is with respect to the radial coordinate, and that we still have to integrate over the angles

Check the normalization;

OK, ψ is normalized.

Taking the second derivative of the radial coordinate

the conditional holds true, so

First, the Coulombic interaction between the nuclei and the electron

    Moving on to the 2 electron effective potential part.  

Now, collect everything together:

just a quick change of variables

and set ξ=2 for the He atom

c.  Now, set in the equation you derived for ε(ζ) above, Plot the ε(ζ) (for He) as a function of these new parameters and determine the best ζ which minimizes the single particle energy ε(ζ).  Compare this value to the table of screening constants in Table 4.9 of Karplus and Porter.

d. This problem is an example of the SCF procedure commonly used to determine the electronic structure of atoms and molecules.  According to Koopman's theorem, the one electron energy of the highest occupied orbital as obtained  via a SCF calculation is a reasonable approximation to the ionization potential (-IP).  How does your estimate for the single particle energy (-ε) compare to the first IP of He (24.39 eV).

The single particle energy predicted here is -0.945313 hartree yielding an IP = 25.807  eV. All in all not too bad!  &:  
The screening given in A&M is a bit higher than the 1.375 predicted here.

The expression used to decompose the electron/electron integral into two parts comes from the expansion of in terms of the spherical harmonics

where

and the notation and are such that = if < ,  c.f Quantum Theory of Angular Momentum by D. A. Varshalovich, A. N. Moskalev, and V. K. Khersohskii for details.